Necessary and Sufficient Quantum Information Characterization Of

Necessary and suﬃcient quantum information characterization of Einstein-Podolsky-Rosen steering

Marco Piani1, 2 and John Watrous3, 4 1Institute for Quantum Computing & Department of Physics and Astronomy, University of Waterloo, 200 University Avenue West, Waterloo, Ontario N2L 3G1, Canada 2Department of Physics and SUPA, University of Strathclyde, Glasgow G4 0NG, UK 3Institute for Quantum Computing & School of Computer Science, University of Waterloo, 200 University Avenue West, Waterloo, Ontario N2L 3G1, Canada 4Canadian Institute for Advanced Research, 180 Dundas Street West, Suite 1400, Toronto, Ontario M5G 1Z8, Canada Steering is the entanglement-based quantum effect that embodies the “spooky action at a distance” disliked by Einstein and scrutinized by Einstein, Podolsky, and Rosen. Here we provide a necessary and sufficient characterization of steering, based on a quantum information processing task: the discrimination of branches in a quantum evolution, which we dub subchannel discrimination. We prove that, for any bipartite steerable state, there are instances of the quantum subchannel discrimination problem for which this state allows a correct discrimination with strictly higher probability than in the absence of entanglement, even when measurements are restricted to local measurements aided by one-way communication. On the other hand, unsteerable states are useless in such conditions, even when entangled. We also prove that the above steering advantage can be exactly quantified in terms of the steering robustness, which is a natural measure of the steerability exhibited by the state.

PACS numbers: 03.67.Mn, 03.67.Bg, 03.65.Ud

Entanglement is a property of distributed quantum for?” can arguably be considered limited [9, 12]. Further- systems that does not have a classical counterpart and more, the quantification of steering has just started to be challenges our everyday-life intuition about the physi- addressed [24]. cal world [1]. It is also the key element in many quan- In this Letter we fully characterize and quantify steer- tum information processing tasks [2]. The strongest fea- ing in an operational way that mirrors the asymmetric ture exhibited by entangled systems is non-locality [3]. features of steering, and that breaks new ground in the A weaker feature related to entanglement is steering: investigation of the usefulness of steering. We prove that roughly speaking, in quantum steering one party can in- every steerable state is a resource in a quantum infor- duce very different ensembles for the local state of the mation task that we dub subchannel discrimination, in other party, beyond what is possible based only on a con- a practically relevant scenario where measurements can ceivable classical knowledge about the other party’s “hid- only be performed locally. Subchannel discrimination is den state” [4, 5]. Steering embodies the “spooky action at the identification of which branch of a quantum evolu- a distance”—in the words of Einstein [6]—identified by tion a quantum system undergoes (see Fig. 1). It is well Schrödinger[7], scrutinized by Einstein, Podolsky, and known that entanglement between a probe and an an- Rosen [8], and formally put on sound ground in [4, 5]. cilla can help in discriminating different channels [31–44]. Not all entangled states are steerable, and not all steer- In [45] it was proven that every entangled state is useful able states exhibit nonlocality [4, 5], but states that ex- in some instance of the subchannel discrimination prob- hibit steering allow for the verification of their entangle- lem. Ref. [46] analyzed the question of whether such an ment in a semi-device independent way: there is no need advantage is preserved when joint measurements on the to trust the devices used by the steering party [4, 5, 9]. output probe and the ancilla are not possible. Here we Besides its foundational interest, steering is interesting prove that, when only local measurements coordinated in practice in bipartite tasks, like quantum key distribu- by forward classical communication are possible, every tion (QKD) [10], where it is convenient or appropriate steerable state remains useful, while non-steerable en- to trust the devices of one of two parties, but not neces- tangled states become useless. We further prove that this sarily of the other one. For example, by exploiting steer- usefulness, optimized over all instances of the subchannel ing, key rates unachievable in a fully device-independent discrimination problem, is exactly equal to the robustness approach [11] are possible, still assuming less about the of steering, a natural way of quantifying steering using devices than in a standard QKD approach [12]. For these techniques similar to the ones used in [24], but based on reasons, steering has recently attracted significant in- the notion of robustness [47–50]. terest, both theoretically and experimentally [13–30], Preliminaries: entanglement and steering.— We will mostly directed to the verification of steering. Nonethe- denote using a ˆ (hat) mathematical entities that are less, an answer to the question “What is steering useful “normalized.” So, for example, a positive semidefinite op- 2

en- erator with unit trace is a (normalized) stateρ ˆ. An Λ1[ρ] Λ1 semble E = {ρa}a for a stateρ ˆ is a collection of substates ρ ≤ ρˆ such that P ρ =ρ ˆ. Each substate ρ is pro- a a a a Λ2 Λ2[ρ] portional to a normalized stateρ â, ρa = paρâ, with . ρ . . pa = Tr(ρa) the probability ofρ â in the ensemble. An as- . Λa semblage A = {Ex}x = {ρa x}a,x is a collection of ensem- | P . Λ [ρ] bles Ex for the same stateρ ˆ, one for each x, i.e., a ρa x = . a | Λˆ . ρˆ, for all x. For example, E = { 1 |0ih0|, 1 |1ih1|} and . ˆ 2 2 √ . Λ[ρ] 1 1 E0 = { 2 |+ih+|, 2 |−ih−|}, with |±i := (|0i ± |1i)/ 2, are both ensembles for the maximally mixed state 11/2 FIG. 1. A decomposition of a channel into subchannels can be of a qubit, and taken together they form an assemblage seen as a decomposition of a quantum evolution into branches ˆ A = {E, E0} for 11/2. Along similar lines, a measurement of the evolution. If {Λa}a is an instrument for Λ, then we can ˆ assemblage MA = {Ma x}a,x is a collection of positive imagine that the evolution ρ 7→ Λ[ρ] has branches ρ 7→ Λa[ρ], | P where each branch takes place with probability Tr(Λa[ρ]). The operators Ma x ≥ 0 satisfying a Ma x = 11 for each x, | | ˆ which thus represents one positive-operator-valued mea- transformation described by the total channel Λ can be seen sure (or POVM), describing a quantum measurement, for as the situation where the “which-branch” information is lost. An example of subchannel discrimination problem is that of each x. For a fixed bipartite stateρ ÂB, every measure- distinguishing between the two quantum evolutions Λ [ρ] = √ i ment assemblage on Alice leads to an assemblage on Bob K ρK†, i = 0, 1, with K = |0ih0| + 1 − γ|1ih1| and K = √i i 0 1 via γ|0ih1|, corresponding to the so-called amplitude damping ˆ B A channel Λ = Λ0 + Λ1 [2]. ρa x = TrA Ma xρÂB . (1) | |

On the other hand, every assemblage on Bob {σa x}a,x sep P | one has TrA(Ma xσAB) = λ p(λ)p(a|x, λ)σB(λ), with has a quantum realization (1) for someρ ÂB satisfying | p(a|x, λ) = TrA(Ma xσA(λ)). It follows that entangle- P | ρˆB = TrA(ˆρAB) = x σa x =:σ ˆB and for some measure- | ment is a necessary condition for steerability, and, in ment assemblage [51]. turn, a steerable assemblage is a clear signature of en- An assemblage A = {ρa x}a,x is unsteerable if | tanglement. Interestingly, not all entangled states lead to steerable assemblages by the action of appropriate local US X X ρa x = p(λ)p(a|x, λ)ˆσ(λ) = p(a|x, λ)σ(λ), (2) measurement assemblages [4, 5]; we call steerable states | λ λ those that do, and unsteerable states those that do not. There exist entangled states that are steerable by one for all a, x, for some probability distribution p(λ), condi- party but not the other (see, e.g., [22]). In this Letter, tional probability distributions p(a|x, λ), and statesσ ˆ(λ). when we refer to a state being steerable or unsteerable, Here λ indicates a (hidden) classical random variable, it is always to be assumed that Alice is the steering party. and we also introduced subnormalized states σ(λ) = Channel and subchannel identification.— A subchannel p(λ)ˆσ(λ). We observe that every conditional probability Λ is a linear completely positive map that is trace non- distribution p(a|x, λ) can be written as a convex combi- increasing: Tr(Λ[ρ]) ≤ Tr(ρ), for all states ρ. If a sub- nation of deterministic conditional probability distribu- channel Λ is trace-preserving, Tr(Λ[ρ]) = Tr(ρ), for all tions: p(a|x, λ) = P p(ν|λ)D(a|x, ν), where D(a|x, ν) = ν ρ, we use the ˆnotation and say that Λˆ is a channel. An δa,fν (x) is a deterministic response function labeled by ν. instrument I = {Λa}a for a channel Λˆ is a collection of This means that, by a suitable relabeling, ˆ P subchannels Λa such that Λ = a Λa (see Figure 1). Ev- US X ery instrument has a physical realization, where the index ρa x = D(a|x, λ)σ(λ) ∀a, x, (3) | a can be considered available to some party [2, 53, 54]. λ Fix an instrument {Λa}a for a channel Λ,ˆ and con- where the summation is over labels of deterministic re- sider a measurement {Qb}b on the output space of Λ.ˆ The sponse functions. We say that an assemblage {ρa x}a,x is joint probability of Λa and Qb for input ρ is p(a, b) := | steerable if it is not unsteerable. Tr(QbΛa[ρ]) = p(b|a)p(a), where p(a) = Tr(Λa[ρ]) is the A separable (or unentangled) state decomposes as probability of the subchannel Λa for the given input ρ sep P σÂB = λ p(λ)ˆσA(λ) ⊗ σˆB(λ), forσ Â(λ),σ ˆB(λ) local and p(b|a) = p(a, b)/p(a) is the conditional probability of states, λ a classical label, and p(λ) a probability distri- the outcome b given that the subchannel Λa took place. bution [52]. A state is entangled if it is not separable. An The probability of correctly identifying which subchannel unsteerable assemblage can always be obtained via (1) was realized is P from the separable state ρAB = λ p(λ)|λihλ|A ⊗σˆ(λ)B, X P pcorr({Λa}a, {Qb}b, ρ) = Tr(QaΛa[ρ]). (4) with Ma x = µ p(a|x, µ)|µihµ|, and hµ|λi = δµλ. Most importantly,| any separable state can only lead a to unsteerable assemblages, as, for a separable state, The archetypal case of subchannel discrimination is 3

E NE ρ Λa Qb nels {Λa}a if pcorr({Λa}a) > pcorr({Λa}a). It is known that there are instances of subchannel discrimination, al- (a) 1 ˆ 1 ˆ ready in the simple setting {Λa}a = { 2 Λ0, 2 Λ1}, where E NE pcorr ≈ 1 pcorr ≈ 0 (see [46] and references therein). Λa B In [45] it was proven that, for any entangled state ρAB, 1 ˆ 1 ˆ there exists a choice { 2 Λ0, 2 Λ1} such that ρAB Qb A n1 1 o n1 1 o (b) p Λˆ , Λˆ , ρ > pNE Λˆ , Λˆ , corr 2 0 2 1 AB corr 2 0 2 1

Λa Nx i.e., that every entangled state is useful for the task of B (sub)channel discrimination. In this sense, every entan- x Qb gled state, independently of how weakly entangled it is,

ρAB Mb x is a resource. Nonetheless, exploiting such a resource may A | require arbitrary joint measurements on the output probe (c) and ancilla [46]. From a conceptual perspective, one may want to limit measurements to those performed by local FIG. 2. Different strategies for subchannel discrimination. (a) operations and classical communication (LOCC), as this No entanglement is used: a probe, initially in the state ρ, un- makes the input entangled state the only non-local re- ˆ dergoes the quantum evolution Λ, with branches Λa, and is source. This limitation can be justified also from a practi- later measured, with an outcome b for the measurement de- cal perspective: LOCC measurements are arguably easier scribed by the POVM {Qb}b, which is the guess for which to implement, and might be the only feasible kind of mea- branch of the evolution actually took place. (b) The probe B is potentially entangled with an ancilla A; the output probe surements, especially in a scenario where only weakly en- and the ancilla are jointly measured. (c) The probe is still tangled states can be produced. We do not know whether potentially entangled with an ancilla, but the final measure- every entangled state stays useful for subchannel discrim- ment {Qb}b is restricted to local measurements on the output ination when measurements are restricted to be LOCC, probe and the ancilla, coordinated by one-way classical com- but we will see that, if the measurements are limited munication (single lines represent quantum systems, double to local operations and forward communication (one-way lines classical information): the outcome x of the measure- LOCC), then only steerable states remain useful. ment {N } performed on the output probe is used to decide x x Steerability and subchannel identification by means which measurement {Mb|x}b to perform on the ancilla. of restricted measurements.— A Bob-to-Alice one-way B A B A LOCC measurement M → = {Qa → }a has the struc- B A P A B B ture Qa → = x Ma x ⊗ Nx , where {Nx }x is a mea- that of channel discrimination, where Λ = p Λˆ , with | a a a surement on B and {M A } is a measurement as- channels Λˆ and probabilities p . The problem often a x a,x a a semblage on A. The optimal| probability of success in considered is that of telling apart just two channels the discrimination of the instrument IB = {ΛB} by Λˆ and Λˆ , each given with probability p = p = a a 0 1 0 1 means of the input state ρ and one-way LOCC mea- 1/2. In this case the total (average) channel is simply AB ˆ 1 ˆ 1 ˆ surements from B to A (see Fig. 2(c)) is given by Λ = 2 Λ0 + 2 Λ1. The best success probability in iden- B A B B A pcorr→ (I, ρAB) := max B→A pcorr(I , M → , ρAB). We tifying subchannels {Λa}a with an input ρ is defined M say that ρAB is useful in this restricted-measurement sce- as pcorr({Λa}a, ρ) := max Qb b pcorr({Λa}a, {Qb}b, ρ). B A NE { } nario if p (I, ρ ) > p (I) for some instrument I Optimizing also over the input state, one arrives at corr→ AB corr NE [55]. Using (1), we find that pcorr({Λa}a) := maxρ pcorr({Λa}a, ρ), where the super- script NE stands for “no entanglement” (see Fig. 2(a)). B B A X B B pcorr(I , M → , ρAB) = TrB(Λa† [Nx ]ρa x), (5) | Indeed, one may try to improve the success probability a,x by using an entangled input state ρAB of an input probe B and an ancilla A. The guess about which subchannel where Λa† denotes the dual map to Λa, which may be took place is based on a joint measurement of the out- defined via Tr(XΛa[Y ]) = Tr(Λa† [X]Y ), ∀X,Y . If the as- put probe and the ancilla (see Fig. 2(b)), with success semblage A = {ρa x}a,x appearing in (5) is unsteerable, B AB | probability pcorr({Λa }a, {Qb }b, ρAB). In the latter ex- then we can achieve an equal or better performance us- pression we have explicitly indicated that the subchan- ing an uncorrelated probe in the best input stateσ ˆ(λ) nels act non-trivially only on B, while input state and among the ones appearing in Eq. (2). Thus, if ρAB is un- measurement pertain to AB. One can define the opti- steerable, then it is useless for subchannel discrimination mal probability of success for a scheme that uses input with one-way measurements. This applies also to entan- E entanglement and global measurements: pcorr({Λa}a) := gled states that are unsteerable, which are nonetheless B AB maxρ max AB pcorr({Λ }a, {Q }b, ρAB). We say useful in channel discrimination with arbitrary measure- AB Qb b a b that entanglement{ } is useful in discriminating subchan- ments [45]. 4

We will now prove that every steerable state is useful Here ◦ is composition, and ΠX indicates the projector in subchannel discrimination with one-way LOCC mea- onto an orthonormal basis {|xi}, x = 1,..., |X|, where surements. To state our result in full detail we need to |X| is the number of settings in the measurement assem- introduce the steering robustness of ρAB, blage MA. The constant α > 0 will be chosen soon. By the conditions (8c), (9), and (10), each Λa† is completely A B Rsteer→ (ρAB) := sup R(A), (6) positive, and therefore so is each Λa; these maps act as P MA Λa[ρ] = α x Tr(Fa xρ)|xihx|, and are subchannels as P P| P where the supremum is over all measurement assemblages long as a Λa† [11] = a,x Λa† [|xihx|] = α a,x Fa x ≤ 11, P | 1 MA = {Ma x}a,x on A, R(A) is the steering robustness a condition that can be satisfied for α = k a,x Fa xk− , of the assemblage| A, defined as the minimum value of with k · k the operator norm. | ∞ ∞ t ≥ 0 for which there exists an assemblage {τa x} for Finally, we introduce N additional subchannels | 1 P which {(ρa x + t τa x)/(1 + t)}a,x is unsteerable, and A Λa[ρ] = Tr((11 − Λa† [11])ρ)ˆσa, for a = |A| + | | N a is obtained from ρAB with the measurement assemblage 1,..., |A| + N, where |A| indicates the original num- MA on A (see Eq. (1)). The steering robustness of A, ber of outcomes for POVMs in MA, andσ â are ar- which is nonzero if and only if A is steerable, is a measure bitrary states in a two-dimensional space orthogonal of the minimal “noise” needed to destroy the steerability to span{|xi | x = 1,..., |X|}. In totality, the sub- of the assemblage A, with noise intended as the mixing channels {Λa} define an instrument I for the trace- P A +N with an arbitrary assemblage {τa x}a,x. We prove the fol- preserving channel Λˆ = | | Λ , and one can in- | a=1 a lowing. corporate the measurement assemblage MA into a one- B A way LOCC strategy M → such that α(1 + R(A)) ≤ Theorem 1. Every steerable state is useful in one-way B B A 2 pcorr(I , M → , ρAB) ≤ α(1+R(A))+ N . On the other subchannel discrimination. More precisely, it holds NE 2 hand, condition (8b) implies α ≤ pcorr(I) ≤ α + N , so B A B B A NE 1+R( ) p (I, ρ ) pcorr(I , M → , ρAB)/pcorr(I) ≥ A . The claim sup corr→ AB = RA B(ρ ) + 1, (7) 1+2/(αN) NE steer→ AB follows by taking N to be arbitrarily large. pcorr(I) I where the supremum is over all instruments I. Conclusions.— We have proven that the steerable states are precisely the states useful for the task of Proof. Details of the proof appear in [56]; a summary is subchannel discrimination with feed-forward local mea- as follows. Using the definitions above, one checks that surements. This answers a question left open by [46] about the characterization of a large class of entan- B B A A B NE pcorr(I , M → , ρAB) ≤ (1 + Rsteer→ (ρAB))pcorr(I), gled states that remain useful for (sub)channel discrimination with local measurements. Most importantly, it B A for any M → and any I. It remains to prove that the provides a full operational characterization—and proof bound can be approximated arbitrarily well by construct- of usefulness—of steering in terms of a fundamental ing appropriate instances of the subchannel discrimina- task, subchannel discrimination, in a setting—that of re- tion problem. To do this, we will need that the steering stricted measurements—very relevant from the practical robustness R(A) of any assemblage A = {ρa x}a,x can be | point of view. The construction in the proof of Theorem 1 calculated via semidefinite programming (SDP) [57]. In proves that, for any measurement assemblage MA on A particular, R(A) + 1 is equal to the optimal value of the such that the corresponding A exhibit steering with ro- SDP optimization problem bustness R(A) > 0, there exist instances of the subchan- X nel discrimination problem with restricted measurements maximize Tr(Fa xρa x) (8a) | | where the use of the steerable state ensures a probability a,x of success approximately (1 + R(A))-fold higher than in X subject to D(a|x, λ)Fa x ≤ 11 ∀λ (8b) the case where no entanglement is used. Thus, the robust- | A B a,x nesses R(A) and Rsteer→ (ρAB) have operational meanings not only in terms of the resilience of steerability versus Fa x ≥ 0 ∀a, x, (8c) | noise, but also in applicable terms. Also, they constitute where each λ labels a deterministic response function. semi-device-independent lower bounds Now, let MA = {Ma x}a,x be a measurement assem- | A B blage on A, and A the resulting assemblage on B. 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[48] M. Steiner, Phys. Rev. A 67, 054305 (2003). as well choose the best input to begin with. So, the only [49] A. Harrow and M. Nielsen, Phys. Rev. A 68, 012308 one-way communication that may have a non-trivial ef- (2003). fect is that from the output probe to the ancilla. [50] J. Geller and M. Piani, arXiv preprint arXiv:1401.8197 [56] See Supplemental Material at [URL will be inserted by (2014). publisher] for details. [51] L. P. Hughston, R. Jozsa, and W. K. Wootters, Physics [57] S. P. Boyd and L. Vandenberghe, Convex optimization Letters A 183, 14 (1993). (Cambridge University Press, 2004). [52] R. F. Werner, Phys. Rev. A 40, 4277 (1989). [58] F. G. S. L. Brandão and N. Datta, IEEE Transactions [53] E. B. Davies and J. T. Lewis, Communications in Math- on Information Theory 57, 1754 (2011). ematical Physics 17, 239 (1970). [59] F. G. S. L. Brandão, Phys. Rev. A 76, 030301(R) (2007). [54] M. Horodecki, Quantum Information and Computation [60] See Supplemental Material [url], which includes Refs. 1, 3 (2001). [61–64]. [55] Notice that no bipartite state ρAB is useful in one-way [61] G. Vidal and R. F. Werner, Phys. Rev. A 65, 032314 subchannel identification when the communication goes (2002). from the ancilla to the output probe. This is because the [62] Z. Jiang, M. Piani, and C. M. Caves, Quantum Informa- initial measurement of the ancilla simply creates an en- tion Processing 12, 1999 (2013). semble of input substates for the channel, and we might [63] I. Bengtsson, arXiv preprint quant-ph/0610216 (2006). [64] R. Jozsa and J. Schlienz, Phys. Rev. A 62, 012301 (2000). Supplemental Material for “Necessary and sufficient quantum information characterization of Einstein-Podolsky-Rosen steering”

ROBUSTNESS AS SEMIDEFINITE PROGRAM can be characterized as the solution to the following optimization problem: Inspired by the work of Pusey [1] and Skrzypczyk et X al. [2], we prove that calculating the steering robustness minimize Tr(σλ) R(A) of an assemblage A = {ρ } falls under the λ a|x a,x X umbrella of semideﬁnite programming (SDP) [3]. subject to D(a|x, λ)σλ ≥ ρa|x ∀a, x (4) The steering robustness of A is deﬁned as λ σ ≥ 0 ∀λ λ nρa|x + t τa|x o R(A) := min t ≥ 0 unsteerable, 1 + t a,x This is an example of SDP optimization problem [3]. For our purposes, the primal problem of an SDP is an opti- {τa|x} an assemblage , (1) mization problem cast as

minimize hC,Xi and therefore corresponds to the minimum nonnegative t such that subject to Φ[X] ≥ B X ≥ 0, US ρa|x = (1 + t)σa|x − tτa|x, ∀a, x, where: US with {σ }a,x an unsteerable assemblage and {τa|x}a,x a|x •h C,Xi is the objective function; an arbitrary assemblage. Notice that, because {ρa|x}a,x US and {σa|x}a,x are assemblages, • B and C are given Hermitian matrices;

US • X is the matrix variable on which to optimize; (1 + t)σa|x − ρa|x τa|x = t •h X,Y i := Tr(X†Y ) is the Hilbert-Schmidt inner also deﬁnes an assemblage, provided that product;

US • Φ is a given Hermiticity-preserving linear map. (1 + t)σa|x ≥ ρa|x, ∀a, x. (2) The dual problem provides a lower bound to the objective US As {σa|x}a,x is unsteerable, function of the primal problem. The dual problem is given by US X σa|x = D(a|x, λ)σ(λ) ∀a, x, (3) λ maximize hB,Y i subject to Φ†[Y ] ≤ C we can rewrite Eq. (2) as the condition Y ≥ 0, X (1 + t) D(a|x, λ)σλ ≥ ρ , ∀a, x, a|x where Φ† is the dual of Φ with respect to the Hilbert- λ Schmidt inner product, and Y is another matrix variable. where each σλ is a subnormalized state, and the sum is One says that strong duality holds when the optimal over all the deterministic strategies to output a given x. values of the primal and dual problems coincide. Strong Considering that the factor (1 + t) can be absorbed into duality holds in many cases, and in particular under the each operator σλ (so that they are generally unnormal- Slater conditions that (i) the primal and dual problems ized, rather subnormalized), one realizes that R(A) + 1 are both feasible, and moreover the primal problem is 2 strictly feasible, meaning that there is a positive deﬁnite and the deﬁnitions

X > 0 such that Φ[X] > B, or (ii) the primal and dual A→B Rsteer (ρAB) := sup R(A), (8) problems are both feasible, and moreover the dual prob- MA lem is strictly feasible, meaning that there is a Y > 0 such that Φ†[Y ] < C. In case (i), not only do the primal and (1): and dual values coincide, but there must exist Y that opt B B→A X †B B achieves the optimal value for the dual problem; and sim- pcorr(I , M , ρAB) = TrB(Λa [Nx ]ρa|x) a,x ilarly, in the case (ii), there must exist Xopt that achieves the optimal value in the primal problem. X †B B US ≤ (1 + R(A)) TrB(Λa [Nx ]σa|x) In our case, C = 11, B = diag(ρa|x)a,x, and a,x X ! − R(A) Tr (Λ†B[N B]τ US ) X B a x a|x Φ[X] = diag D(a|x, λ)Xλ , a,x λ a,x NE A→B NE ≤ (1 + R(A))pcorr(I) ≤ (1 + Rsteer (ρAB))pcorr(I). where diag(·)a,x indicates a block-diagonal matrix whose On the other hand, suppose that MA = {Ma|x}a,x, diagonal blocks are labeled by a, x, and the Xλ operators where a = 1,..., |A| and x = 1,..., |X|, is a measure- are the diagonal blocks of X, labeled by λ. Thus, we have ment assemblage on A such that the corresponding as- A ! semblage A = {ρa|x = TrA(Ma|xρAB)}a,x is steerable. † X Φ [Y ] = diag D(a|x, λ)Ya|x , Let Fa|x ≥ 0 be the operators optimal for (5), such that a,x P λ a,x Tr(Fa|xρa|x) = 1+R(A). In the proof of Theorem 1 of the main text we deﬁned subchannels Λ that act as and the dual of the primal problem (4) reads a

X Λa[ρ] = maximize Tr(Fa|xρa|x) (5a) a,x ( P|X| α x=1 Tr(ρFa|x)|xihx| 1 ≤ a ≤ |A| X 1 P|A| † subject to D(a|x, λ)Fa|x ≤ 11 ∀λ (5b) N Tr((11 − a=1 Λa[11])ρ)ˆσa |A| + 1 ≤ a ≤ |A| + N, a,x (9) F ≥ 0 ∀a, x, (5c) a|x P −1 where α = k a,x Fa|xk∞ > 0, and theσ â, It is easy to verify that both Slater conditions hold a = |A| + 1,..., |A| + N, are arbitrary (normalized) in our case. For instance, one can take σλ = 211 for states in a two-dimensional subspace orthogonal to 11 span{|xi | x = 1,..., |X|}. It is immediate to check all λ, and Fa|x = |X|+1 for all a, x, with |X| being P|A|+N the number of possible values for x. Thus, there exist that Tr Λa[ρ] = Tr(ρ) (by construction), so opt a=1 Fa|x = Fa|x satisfying the constraints of Eq. (5) and I = {Λ } is an instrument for the channel P a a=1,...,|A|+N such that a,x Tr(Fa|xρa|x) = 1 + R(A). P|A|+N a=1 Λa. We remark that the optimal Fa|x can always be chosen Let σAB be an arbitrary bipartite state on AB, and to saturate (5b). That is, there is a deterministic strategy B→A B→A let MA = {Qa}a be an arbitrary one-way mea- D(a|x, λ) and a normalized pure state |φi such that B→A P 0A 0B surement from B to A, i.e., Qa = y Ma|y ⊗ Ny , X to guess which subchannel was actually realized. Notice D(a|x, λ)hφ|Fa|x|φi = hφ|11|φi = 1 (6) a,x that y in the latter expression potentially varies in an arbitrary range, different from the range {1,..., |X|} for This is because otherwise it is always possible to increase the parameter x of the fixed measurement assemblage (in operator sense) some F ’s, still maintaining the op- 0 a|x MA. Nonetheless we observe that Λa = ΠX ◦ Λa for timal value for the objective function (which is operator a = 1,..., |A| + N, where ◦ is composition, and monotone in the Fa|x’s). |X| 0 X ⊥ ⊥ ΠX [τ] = |xihx|τ|xihx| + Π τΠ , DETAILS OF THE PROOF OF THEOREM 1 x=1 with Π⊥ the projector onto the two-dimensional space The claimed upper bound, orthogonal to span{|xi | x = 1,..., |X|} that supports B B→A A→B NE the arbitrary qubits statesσ â, a = |A| + 1,..., |A| + N. pcorr(I , M , ρAB) ≤ (1 + R (ρAB))p (I), steer corr Also, can be proved using  |A|  1 B B→A X †B B B X B B p (I , M , ρ ) = Tr (Λ [N ]ρ ), (7) Λ [σAB] = σA − TrB(Λ 0 [σAB]) ⊗ σˆ , corr AB B a x a|x a N  a  a a,x a0=1 3 for a = |A| + 1,..., |A| + N. This implies that, for what- therefore ever input σ , the optimal QB→A can be chosen to have AB a p (IB, MB→A, ρ ) the form corr AB |A| |X| B→A X X Qa = α Tr(Fa|xσa|x) ( a=1 x=1 P|X| M 0A ⊗ |xihx|B 1 ≤ a ≤ |A| = x=1 a|x  |A|  |A|+N A B X 1 X 11 ⊗ Na , |A| + 1 ≤ a ≤ |A| + N, + 1 − Tr(ΛB[σ ]) Tr(N σˆ )  a AB  N a a (10) a=1 a=|A|+1 with Π⊥N Π⊥ = N , for |A| + 1 ≤ a ≤ |A| + N, a |A| |X| a a X X 2 POVM on the orthogonal qubit space. Omitting a de- ≤ α Tr(F σ ) + . a|x a|x N tailed and straightforward proof of this, we instead pro- a=1 x=1 vide the following intuition: For the subchannels (9), the (11) best local measurement on the output probe is one that In the last line we used first of all discriminates between the space span{|xi | x =  |A|  X B 1,..., |X|} and the orthogonal qubit space. If the probe 1 − Tr(Λa [σAB]) ≤ 1 is found in the space span{|xi | x = 1,..., |X|}, the probe a=1 is then measured in the basis {|xi | x = 1,..., |X|} and the result if forwarded to decide which measurement to and perform on the ancilla: this is optimal because, in this |A|+N  |A|+N  subspace, the output probe is already dephased in the 1 X 1 X Tr(Naσâ) ≤ Tr  Na basis {|xi | x = 1,..., |X|}. If the probe is instead found N N a=|A|+1 a=|A|+1 in the orthogonal qubit space, there is no information 1 2 to be gained from the ancilla, since, for the state of the ≤ Tr(Π⊥) = . (12) probe to have support in the orthogonal qubit space, the N N 0 probe must have been discarded and prepared in one of It is clear that if σAB = ρAB and Ma|x = Ma|x in (10), the random qubit statesσ â. So, in this case, the ancilla is so that σa|x = ρa|x, then we have necessarily decorrelated and its state independent of the 2 specific Λ , a = |A| + 1,..., |A| + N, that has been real- 1 + R(A) ≤ p (IB, MB→A, ρ ) ≤ 1 + R(A) + . a corr AB N ized; thus the optimal guess about said Λa can be made as soon as the output probe is measured. It remains to prove that B→A B→A Then, for an optimal M = {Qa }a of the form 2 α ≤ pNE (I) ≤ α + . (13) (10), we find in general corr N B B→A pcorr(I , M , σAB) This is readily verified by considering that (5b) can be |A|+N saturated, as argued at the end of the previous section X B→A B = Tr(Qa Λa [σAB]) (see (6)), for an optimal solution of the SDP problem. So a=1 we have that for some deterministic D(a|x, λ) and some |A| |A|+N uncorrelated input state |φi to the channel, X X = Tr(QB→AΛB[σ ]) + Tr(QB→AΛB[σ ]) a a AB a a AB |A| |X| a=1 a=|A|+1 X X 1 = D(a|x, λ)hφ|Fa|x|φi |A| |X| a=1 x=1 X X 0A B = Tr(Ma|x ⊗ |xihx|BΛa [σAB]) |A| |X| 1 X X a=1 x=1 = D(a|x, λ)hφ|Λ† [|xihx|]|φi   α a |A| |A|+N a=1 x=1 X B 1 X + 1 − Tr(Λa [σAB]) Tr(Naσâ) |A|    N 1 X X a=1 a=|A|+1 = Tr |xihx| Λ [|φihφ|] α   a  |A| |X| a=1 x:D(a|x,λ)=1 X X † = Tr(Λa[|xihx|]σa|x) |A| 1 X a=1 x=1 = Tr (M 00Λ [|φihφ|]) ,   α a a |A| |A|+N a=1 X B 1 X + 1 − Tr(Λa [σAB]) Tr(Naσâ), 00 P N having defined Ma := |xihx|. Considering a=1 a=|A|+1 x:D(a|x,λ)=1 also the subchannels Λa, a = |A| + 1,..., |A| + N, and 0 with σa|x = TrA(Ma|xσAB). By construction it holds that bounding their contribution to the probability of success † Λa[|xihx|] = αFa|x for 1 ≤ a ≤ |A| and 1 ≤ x ≤ |X|, as in (11), we arrive at (13). 4

ON THE SCALING OF THE STEERABILITY OF of the coefficients of |ψa|xi in the local basis {|iiA}. Thus, MAXIMALLY ENTANGLED STATES ∗ the bases {|ψa|xi}a=1,...,d are still mutually unbiased. We want to lower bound the steering robustness of A→B B We have argued that Rsteer (ρAB) ≤ Rg(ρAB), where {ρa|x}a,x, which in turn will give us a lower bound on R (ρ ) is the generalized entanglement robustness A→B + g AB Rsteer (ψd,AB). To do this, we use a specific choice for the F ’s in (5). We choose F = β|ψ∗ ihψ∗ |, where n ρ + t τ o a|x a|x a|x a|x R (ρ ) = min t ≥ 0 AB AB separable, τ a state . g AB 1 + t β > 0 will be fixed to satisfy (5b) (condition (5c) is satisfied for any β ≥ 0), i.e., Indeed, let τAB be optimal for the generalized entanglement robustness, i.e., suppose X D(a|x, λ)Fa|x ≤ 1 a,x ρAB + Rg(ρAB)τAB ∞ σAB = 1 + Rg(ρAB) for all deterministic D(a|x, λ). With our choice of Fa|x, this can be achieved by taking is separable. Then σa|x = TrA(Ma|xσAB) is unsteerable !−1 for any measurement assemblage {Ma|x}a,x, proving that A→B X ∗ ∗ Rg(ρAB) is an upper bound to R (ρAB) (see Eq. (8)). β ≤ max |ψf (x)|xihψf (x)|x| (14) steer λ λ λ This means that, if a state is weakly entangled with re- x ∞ spect to Rg, it is also weakly steerable with respect to the maximum being over all functions fλ : {1, . . . , d + A→B Rsteer . In [4] it was proven that, for any bipartite pure 1} → {1, . . . , d}, labeled by λ. To estimate the right hand state side of (14), we will use the fact [8] that, for √ X d+1 |ψiAB = pi|iiA|iiB, X |γi = |ψ∗ i |xi , i CD fλ(x)|x C D x=1 here in its Schmidt decomposition, the generalized entanglement robustness is equal to where {|xi}x=1....,d+1 is an orthonormal basis, the spectrum of !2 X ∗ ∗ X √ TrD(|γihγ|CD) = |ψ ihψ |. Rg(|ψihψ|AB) = pi − 1 = 2N (|ψihψ|AB), fλ(x)|x fλ(x)|x x i is the same as the spectrum of where N is the negativity of entanglement [5]. In partic- X ∗ ∗ ular, then, for a maximally entangled state in dimension TrC (|γihγ|CD) = hψf (x)|x|ψf (y)|yi|yihx| + 1 d λ λ d × d, |ψ i = √ P |ii |ii , one has x,y d AB d i=1 A B X 1 X iφ A→B + + = |xihx| + √ e x,y |yihx| Rsteer (ψd,AB) ≤ Rg(ψd,AB) = d − 1, x d x6=y having used the notation ψ+ = |ψ+ihψ+| . 1 1 X d,AB d d AB = 1 − √ 11 + √ eiφx,y |yihx| We conclude by providing a lower bound on d d A→B + x,y Rsteer (ψd,AB) for d a power of a prime number. We will use techniques similar to the ones used in the examples where φx,y are real numbers representing phases. Thus, of [6]. we have Fix d to be the power of a prime number. Then we X know that there there are d + 1 mutually unbiased bases, |ψ∗ ihψ∗ | fλ(x)|x fλ(x)|x i.e., d + 1 orthonormal sets {|ψa|xi}a=1,...,d, one for each x ∞ x = 1, . . . , d + 1, such that [7] X = hψ∗ |ψ∗ i|yihx| ( fλ(x)|x fλ(y)|y δa,b x = y x,y ∞ |hψa|x|ψb|yi| = 1 √ x 6= y 1 1 X d iφx,y ≤ 1 − √ + √ e |yihx| d d x,y We will consider a measurement assemblage {Ma|x = ∞ + |ψa|xihψa|x|}a,x. Suppose ρAB = ψd,AB. We have 1 1 X ≤ 1 − √ + √ eiφx,y |yihx| d d B A + 1 ∗ ∗ x,y 2 ρ = TrA(M ψ ) = |ψ ihψ | a|x a|x d,AB d a|x a|x 1 1 = 1 − √ + √ (d + 1) Here |ψ∗ i indicates orthonormal vectors whose coeffi- d d a|x √ cients in the local basis {|iiB} are the complex conjugate = 1 + d. 5

Since√ this estimate is independent of λ, we can take β = 1/( d + 1). Hence, we conclude that, for d the power of a prime number, [1] M. F. Pusey, Phys. Rev. A 88, 032313 (2013), URL http: //link.aps.org/doi/10.1103/PhysRevA.88.032313. A→B + Rsteer (ψd,AB) [2] P. Skrzypczyk, M. Navascu´es,and D. Cavalcanti, Phys. Rev. Lett. 112, 180404 (2014), URL http://link.aps. 1 ∗ ∗ ≥ R |ψa|xihψa|x| org/doi/10.1103/PhysRevLett.112.180404. d [3] S. P. Boyd and L. Vandenberghe, Convex optimization X 1 1 (Cambridge university press, 2004). ≥ Tr |ψ∗ ihψ∗ | √ |ψ∗ ihψ∗ | − 1 d a|x a|x d + 1 a|x a|x [4] A. Harrow and M. Nielsen, Phys. Rev. A 68, 012308 a,x (2003). 1 [5] G. Vidal and R. F. Werner, Phys. Rev. A 65, = √ (d(d + 1)) − 1 d( d + 1) 032314 (2002), URL http://link.aps.org/doi/10. √ 1103/PhysRevA.65.032314. √ d − 1 [6] Z. Jiang, M. Piani, and C. M. Caves, Quantum informa- = d√ d + 1 tion processing 12, 1999 (2013). √ [7] I. Bengtsson, arXiv preprint quant-ph/0610216 (2006). ≥ d − 2. [8] R. Jozsa and J. Schlienz, Phys. Rev. A 62, 012301 (2000). (15)